Interpolating seismic data

ABSTRACT

A technique includes modeling interpolated seismic measurements as a random process characterized by seismic measurements acquired at a set of sensor locations and an interpolation error. The technique includes determining the interpolated seismic measurements based at least in part on a minimization of the interpolation error.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.12/168,125 filed Jul. 5, 2008, which is incorporated herein by referencein its entirety.

BACKGROUND

The invention generally relates to interpolating seismic data.

Seismic exploration involves surveying subterranean geologicalformations for hydrocarbon deposits. A survey typically involvesdeploying seismic source(s) and seismic sensors at predeterminedlocations. The sources generate seismic waves, which propagate into thegeological formations creating pressure changes and vibrations alongtheir way. Changes in elastic properties of the geological formationscatter the seismic waves, changing their direction of propagation andother properties. Part of the energy emitted by the sources reaches theseismic sensors. Some seismic sensors are sensitive to pressure changes(hydrophones), others to particle motion (e.g., geophones), andindustrial surveys may deploy only one type of sensors or both. Inresponse to the detected seismic events, the sensors generate electricalsignals to produce seismic data. Analysis of the seismic data can thenindicate the presence or absence of probable locations of hydrocarbondeposits.

Some surveys are known as “marine” surveys because they are conducted inmarine environments. However, “marine” surveys may be conducted not onlyin saltwater environments, but also in fresh and brackish waters. In onetype of marine survey, called a “towed-array” survey, an array ofseismic sensor-containing streamers and sources is towed behind a surveyvessel.

In exploration seismology, although the time coordinate is regularly ornon-uniformly, sampled, spatial coordinates are typically irregularly,or non-uniformly, sampled due to the presence of obstacles in the landenvironment or strong currents that are present in the marineenvironment. Thus, for example, for the marine environment, even withthe latest steering technology, it may not be possible to maintain thestreamers parallel to each other. Furthermore, the seismic sensors in agiven streamer may not be equidistantly spaced apart. Hence, the inlinesampling may also be quite non-uniform.

The regularization of seismic data typically is very important,especially in time-lapse survey matching, multiple suppression andimaging. If the irregular nature of the sampling grid is ignored orhandled poorly, notable errors may be introduced, and the errors may befurther amplified at later stages of the seismic data processing chain.

SUMMARY

In an embodiment of the invention, a technique includes modelinginterpolated seismic measurements as a random process that ischaracterized by seismic measurements acquired at sensor locations andan interpolation error. The technique includes determining theinterpolated seismic measurements based at least in part on aminimization of the interpolation error.

In another embodiment of the invention, an article includes instructionsstored on a computer accessible storage medium that when executed by aprocessor-based system cause the processor-based system to modelinterpolated seismic measurements as a random process that ischaracterized by seismic measurements acquired at sensor locations andan interpolation error. The instructions when executed cause theprocessor-based system to determine the interpolated seismicmeasurements based at least in part on a minimization of theinterpolation error.

In yet another embodiment of the invention, a system includes aninterface to receive seismic measurements acquired at non-uniformlyspaced sensor locations. The system includes a processor to modelinterpolated seismic measurements at uniformly spaced locations as arandom process that is characterized by the seismic measurementsacquired at the non-uniformly spaced sensor locations and aninterpolation error. The processor determines the interpolated seismicmeasurements based at least in part on a minimization of theinterpolation error.

Advantages and other features of the invention will become apparent fromthe following drawing, description and claims.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a schematic diagram of a marine acquisition system accordingto an embodiment of the invention.

FIGS. 2 and 3 are flow diagrams depicting techniques to interpolateseismic data according to embodiments of the invention.

FIGS. 4, 5 and 6 are flow diagrams depicting techniques to performautocorrelation of measured seismic data according to embodiments of theinvention.

FIG. 7 is a schematic diagram of a seismic data processing systemaccording to an embodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 depicts an embodiment 10 of a marine seismic data acquisitionsystem in accordance with some embodiments of the invention. In thesystem 10, a survey vessel 20 tows one or more seismic streamers 30 (twoexemplary streamers 30 being depicted in FIG. 1) behind the vessel 20.The seismic streamers 30 may be several thousand meters long and maycontain various support cables (not shown), as well as wiring and/orcircuitry (not shown) that may be used to support communication alongthe streamers 30. In general, each streamer 30 includes a primary cableinto which is mounted seismic sensors that record seismic signals.

In accordance with embodiments of the invention, the seismic sensors aremulti-component seismic sensors 58, each of which is capable ofdetecting a pressure wavefield and at least one component of a particlemotion that is associated with acoustic signals that are proximate tothe multi-component seismic sensor 58. Examples of particle motionsinclude one or more components of a particle displacement, one or morecomponents (inline (x), crossline (y) and vertical (z) components (seeaxes 59, for example)) of a particle velocity and one or more componentsof a particle acceleration.

Depending on the particular embodiment of the invention, themulti-component seismic sensor 58 may include one or more hydrophones,geophones, particle displacement sensors, particle velocity sensors,accelerometers, pressure gradient sensors, or combinations thereof.

For example, in accordance with some embodiments of the invention, aparticular multi-component seismic sensor 58 may include a hydrophone 55for measuring pressure and three orthogonally-aligned accelerometers 50to measure three corresponding orthogonal components of particlevelocity and/or acceleration near the seismic sensor 58. It is notedthat the multi-component seismic sensor 58 may be implemented as asingle device (as depicted in FIG. 1) or may be implemented as aplurality of devices, depending on the particular embodiment of theinvention. A particular multi-component seismic sensor 58 may alsoinclude pressure gradient sensors 56, which constitute another type ofparticle motion sensors. Each pressure gradient sensor measures thechange in the pressure wavefield at a particular point with respect to aparticular direction. For example, one of the pressure gradient sensors56 may acquire seismic data indicative of, at a particular point, thepartial derivative of the pressure wavefield with respect to thecrossline direction, and another one of the pressure gradient sensorsmay acquire, a particular point, seismic data indicative of the pressuredata with respect to the inline direction.

The marine seismic data acquisition system 10 includes one or moreseismic sources 40 (one exemplary source 40 being depicted in FIG. 1),such as air guns and the like. In some embodiments of the invention, theseismic sources 40 may be coupled to, or towed by, the survey vessel 20.Alternatively, in other embodiments of the invention, the seismicsources 40 may operate independently of the survey vessel 20, in thatthe sources 40 may be coupled to other vessels or buoys, as just a fewexamples.

As the seismic streamers 30 are towed behind the survey vessel 20,acoustic signals 42 (an exemplary acoustic signal 42 being depicted inFIG. 1), often referred to as “shots,” are produced by the seismicsources 40 and are directed down through a water column 44 into strata62 and 68 beneath a water bottom surface 24. The acoustic signals 42 arereflected from the various subterranean geological formations, such asan exemplary formation 65 that is depicted in FIG. 1.

The incident acoustic signals 42 that are acquired by the sources 40produce corresponding reflected acoustic signals, or pressure waves 60,which are sensed by the multi-component seismic sensors 58. It is notedthat the pressure waves that are received and sensed by themulti-component seismic sensors 58 include “up going” pressure wavesthat propagate to the sensors 58 without reflection, as well as “downgoing” pressure waves that are produced by reflections of the pressurewaves 60 from an air-water boundary 31.

The multi-component seismic sensors 58 generate signals (digitalsignals, for example), called “traces,” which indicate the acquiredmeasurements of the pressure wavefield and particle motion. The tracesare recorded and may be at least partially processed by a signalprocessing unit 23 that is deployed on the survey vessel 20, inaccordance with some embodiments of the invention. For example, aparticular multi-component seismic sensor 58 may provide a trace, whichcorresponds to a measure of a pressure wavefield by its hydrophone 55;and the sensor 58 may provide one or more traces that correspond to oneor more components of particle motion, which are measured by itsaccelerometers 50.

The goal of the seismic acquisition is to build up an image of a surveyarea for purposes of identifying subterranean geological formations,such as the exemplary geological formation 65. Subsequent analysis ofthe representation may reveal probable locations of hydrocarbon depositsin subterranean geological formations. Depending on the particularembodiment of the invention, portions of the analysis of therepresentation may be performed on the seismic survey vessel 20, such asby the signal processing unit 23. In accordance with other embodimentsof the invention, the representation may be processed by a seismic dataprocessing system (such as an exemplary seismic data processing system320 that is depicted in FIG. 7 and is further described below) that maybe, for example, located on land or on the vessel 20. Thus, manyvariations are possible and are within the scope of the appended claims.

The down going pressure waves create an interference known as “ghost” inthe art. Depending on the incidence angle of the up going wavefield andthe depth of the streamer 30, the interference between the up going anddown going wavefields creates nulls, or notches, in the recordedspectrum. These notches may reduce the useful bandwidth of the spectrumand may limit the possibility of towing the streamers 30 in relativelydeep water (water greater than 20 meters (m), for example).

The technique of decomposing the recorded wavefield into up and downgoing components is often referred to as wavefield separation, or“deghosting.” The particle motion data that is provided by themulti-component seismic sensors 58 allows the recovery of “ghost” freedata, which means data that is indicative of the upgoing wavefield.

Due to wave and possibly other effects, the pressure and particle motionmeasurements typically are acquired at irregularly, or non-uniformly,spaced sensor locations. In other words, the actual sensor locations donot resemble idealized locations on a uniform grid. Therefore, beforethe pressure and particle motion measurements that are acquired by theseismic sensors are processed (for such purposes of survey matching,multiple suppression and/or imaging, as examples), the measurements areinterpolated to produce interpolated measurements at idealized,uniformly spaced locations.

Particle motion and pressure wavefield signals that are continuous inthe inline (x), crossline (y) and depth (z) directions may beconstructed from the sampled pressure and particle motion measurements.The Whittaker-Kotel'nikow-Shannon sampling theorem states that anysignal f(x) can be reconstructed from its uniformly spaced samples ifthe sampling interval is less than half the period of the highestfrequency component in that signal. Thus, if f(x) is bandlimited to thewavenumber σ/2, which is known as the Nyquist wavenumber, then thesampling theorem provides the following formula to interpolate anyfunction value from uniformly spaced values f(m/σ):

$\begin{matrix}{{{f(x)} = {\sum\limits_{m = {- \infty}}^{\infty}{{f\left( {m/\sigma} \right)}\sin \; {c\left( {{\sigma \; x} - m} \right)}}}},} & {{Eq}.\mspace{14mu} 1}\end{matrix}$

where “sin c(x),” in general, refers to the expression sin(πx)/πx. Thus,when the sampling rate is sufficient and there is no aliasing, thesampling theorem provides a way to reconstruct the continuous signal“exactly” from its uniformly spaced samples. To satisfy requirements ofthe sampling theorem, the signal is sampled at a rate greater than twicethe Nyquist rate, i.e., σ. Moreover, the seismic signal may berepresented as follows:

$\begin{matrix}{{{{f(x)} \approx {f_{L}(x)}} = {\sigma {\sum\limits_{m = 0}^{L - 1}{\Delta \; x_{m}{f\left( x_{m} \right)}\sin \; {c\left( {\sigma \; \left( {x - x_{m}} \right)} \right)}}}}},} & {{Eq}.\mspace{14mu} 2}\end{matrix}$

where “Δx_(m)” is the Jacobian weight (i.e., Δx_(m)=x_(m+1)−x_(m)); and“f(x_(m))” the value of the seismic data at an irregulator offset called“x_(m).” It is important to note that, when Δx_(m)=1/σ, the sin cinterpolator of Eq. 2 is exact due to the following relationship that isset forth by the Whittaker-Kotel'kikov-Shannon theorem:

$\begin{matrix}{{f(x)} = {\sum\limits_{m = {- \infty}}^{\infty}{{f\left( {m/\sigma} \right)}\sin \; {{c\left( {{\sigma \; x} - \tau} \right)}.}}}} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

On the other hand, when Δx_(m) is not equal 1/σ, the sin c interpolatorprovides only a crude approximation to the continuous signal. For suchcases, it has been found that a better approach is to not obtain thecontinuous signal, but rather, this other approach involves invertingEq. 1 to obtain the discrete, uniformly spaced signal values f(m/σ).This inversion may be written in matrix notation as follows:

$\begin{matrix}{{h \equiv {\begin{matrix}{f\left( x_{1} \right)} \\{f\left( x_{2} \right)} \\\vdots \\{f\left( x_{L} \right)}\end{matrix}} \approx {{\begin{matrix}s_{11} & s_{11} & \ldots & s_{1L} \\s_{21} & s_{21} & \ldots & s_{2L} \\\vdots & \vdots & \ldots & \vdots \\s_{11} & s_{11} & \ldots & s_{LL}\end{matrix}} \cdot {\begin{matrix}{f(0)} \\{f\left( {1/\sigma} \right)} \\\; \\{f\left( {\left( {L - 1} \right)/\sigma} \right)}\end{matrix}}} \equiv S_{g}},} & {{Eq}.\mspace{14mu} 4}\end{matrix}$

where “σ/2” represents the bandwidth of the signal “f(x);” and “S”represents the sin c matrix with entries s_(ij)=sin c(σ(x_(i)−j/σ)). Ifthe matrix S is well conditioned, then the seismic data at regularoffsets may be computed by standard matrix inversion, as describedbelow:

g=S ⁻¹ h.  Eq. 5

Otherwise, a least squares minimum norm inversion may be used, as setforth below:

g=(S ^(T) W ₁ S+W ₂)⁻¹ S ^(T) W ₁ h,  Eq. 6

where “W₁” represents usually chosen as a diagonal matrix whose “m^(th)”diagonal entry is the Jacobian weight Δx_(m)=x_(m+1)−x_(m); and “W₂”represents usually chosen as a small multiple of identity matrix, i.e.,W₂=ε²I.

An interpolator in accordance with Eq. 6 has been described by J. L. Yenin an article entitled, “Non-Uniform Sampling of Bandwidth-LimitedSignals.” IRE Trans. Circuit Theory, CT-3 251-257 (1956). It issometimes referred to as Yen's Interpolator of Type 1.

Many interpolators that are used in seismic data processing arevariations of the Yen's interpolator of Type 1. For example,interpolators are described in Duijndam, J. J. W., Schonewille, M. A.,Hindriks, C. O. H., “Irregular and Sparse Sampling in ExplorationSeismology,” which is a chapter of L. Zhang, NONUNIFORM SAMPLING: THEORYAND PRACTICE, Kluwer Academic/Plenum Publishers, New York, USA (2001).The regularization in Duijndam is formulated as a spectral domainproblem, and the spectrum of the signal is estimated by taking anon-uniform Fourier transform of the irregular samples. The regularizedsamples are constructed by using inverse discrete Fouriertransformation. It can be shown that this regularization technique isexactly equivalent to Yen's Interpolator of Type 1.

A variant of Duijndam's interpolator is described in Zwartjes, P. M. andM. D. Sacchi, 2004, “Fourier Reconstruction of Non-Uniformly Sampled,Aliased Data,” 74^(th) Ann. Internat. Mtg., Soc. Of Expl. Geophys.(1997-2000). Zwartjes' interpolator involves a least squares inversionof the Fourier transformation instead of using an inverse discreteFourier transform. To this purpose, a cost function is defined, whichalso involves a non-quadratic penalty term to obtain a parsimoniousmodel.

Another interpolator is described in Hale, I. D., “ResamplingIrregularly Sampled Data,” Stanford Exploration Project, SEP-25, 39-58(1980). Hale's interpolator is based on a more general version of Yen'sInterpolator, where a space limited signal assumption is not used. Inthat case, the uniform samples f(m/σ) may be determined by solving amatrix equation similar to Eq. 6. Hale discloses replacing the entriesin the inverse matrix by their locally computed approximations.

The interpolators based on Yen's first theorem usually providesatisfactory results on non-aliased signals that have littlehigh-wavenumber content. However, their performance typically degradessignificantly when the interpolated signal has a substantial amount ofhigh wavenumber spectral content. Another shortcoming of theinterpolators based on Yen's first theorem is that in order to solve Eq.4, at least as many irregular sampling positions as regular samplingpositions are required. Hence, if some seismic traces are dropped out,traces which reside at further locations typically are used to solve thesystem of equations given by Eq. 4. Usually this degrades the accuracyof the interpolated sample values. Further, although Yen's firstinterpolator is exact for infinite length signals, it is anapproximation when only a finite extent of the signal is available forinterpolation.

Another interpolator has been described in Yen, J. L., “Non-UniformSampling of Bandwidth-Limited Signals,” IRE Trans. Circuit Theory, CT-3,251-257 (1956) and in U.S. patent application Ser. No. 12/518,533,entitled, “REGULARISATION OF IRREGULARLY SAMPLED SEISMIC DATA” which wasfiled on Jun. 10, 2009 and is hereby incorporated by reference in itsentirety. This interpolator is sometimes referred to as Yen'sInterpolator of Type 4. Given N=L arbitrary positions of seismicreceivers x_(m) and corresponding signal values f(x_(m)), the seismicsignals can be interpolated to N′=L desired or nominal receiverpositions y_(k) using the following relationship:

$\begin{matrix}{{{h_{L}\left( y_{k} \right)} = {\sum\limits_{m = 1}^{L}{\sum\limits_{n = 1}^{L}{\gamma_{mn}{f\left( x_{n} \right)}{\phi \left( {y_{k},x_{m}} \right)}}}}},} & {{Eq}.\mspace{14mu} 7}\end{matrix}$

where “y_(mn)” represents the (m,n)-th element of the inverse of amatrix Γ. The matrix Γ has as its (i,j)-th element φ(x_(i)x_(j)) withφ(y,x) being equal to sin c(σ(y−x)). The advantage of this interpolatoris that the interpolator minimizes the least squares interpolation errorin the spectral domain in cases where no prior information is availableon signal except for that it is band limited. However, if priorinformation on the data is available, e.g., when data are known to benot full band (Bandwidth=Nyquist), smaller interpolation errors may beobtained by exploiting the prior information.

In accordance with embodiments of the invention described herein, forpurposes of obtaining an interpolated set of seismic data at uniformlyspaced locations, an interpolator is used which models the interpolatedseismic data as a stochastic, or random, process. This random process ischaracterized by the seismic data measurements that have been acquiredat non-uniformly-spaced locations and an interpolation error, asdescribed below:

$\begin{matrix}{{{h(y)} = {{\sum\limits_{m}{\alpha_{m\;}{h\left( x_{m} \right)}}} + {\sum\limits_{m}{\beta_{m}{h^{\prime}\left( x_{m} \right)}}} + {w(y)}}},} & {{Eq}.\mspace{14mu} 8}\end{matrix}$

where “x_(m)” represents to the irregular, or non-uniformly, spacedsensor locations; “h(y)” represents the interpolated measurement at anarbitrary position “y;” “h(x_(m))” represents the pressure datameasurement acquired at the x_(m) location; “α_(m)” represents acoefficient to be determined, as described below; “h′(x_(m))” representsthe pressure gradient data (i.e., the particle motion data measurement)acquired at the x_(m) location; “β_(m)” represents a coefficient to bedetermined, as described below; and “w(y)” represents the interpolationerror. It is noted that the measurements may be two-dimensional (2-D) orthree-dimensional (3-D) measurements, depending on the particularembodiment of the invention.

For purposes of clarity, the problem has been formulated herein forco-located sensor pairs. However, the techniques and systems that aredescribed herein may be generalized to non-co-located sensors. Thus,many variations are contemplated and are within the scope of theappended claims.

The statistically optimal (in the least-squares sense) estimate of h(y)may be obtained by minimizing the variance of the interpolation error,as described below:

$\begin{matrix}{{{{Var}\left\lbrack {w(y)} \right\rbrack} = {E\left\lbrack {{{h(y)} - {\sum\limits_{m}{\alpha_{m}{h\left( x_{m} \right)}}} - {\sum\limits_{m}{\beta_{m}{h^{\prime}\left( x_{m} \right)}}}}}^{2} \right\rbrack}},} & {{Eq}.\mspace{14mu} 9}\end{matrix}$

where “Var[w(y)]” represents the variance of the interpolation error;and “E[ ]” represents the statistical expectation operation, where therandom process is assumed for this example to have a mean of zero.

As a more specific example, if an assumption is made that the randomprocess is wide sense stationary (WSS), then its autocorrelationfunction may be expressed as follows:

E[h(x _(m))h*(x _(n))]=R _(hh)(x _(m) −x _(n)),  Eq. 10

where “*” represents complex conjugation; and “R_(hh)” represents theautocorrelation function. Using this notation, the variation of theinterpolation error may be expressed as follows:

$\begin{matrix}{{{Var}\left\lbrack {w(y)} \right\rbrack} = {{{R_{hh}(0)} - {\sum\limits_{m}{\alpha_{m}{R_{h}\left( {x_{m} - y} \right)}}} - {\sum\limits_{m}{\beta_{m}\frac{\partial{R_{h}\left( {x_{m} - y} \right)}}{\partial x_{m}}}} - {\sum\limits_{m}{\alpha_{m}^{*}{R_{h}\left( {y - x_{n}} \right)}}} + {\sum\limits_{m}{\sum\limits_{n}{\alpha_{m}\alpha_{n}{R_{h}\left( {x_{m} - x_{n}} \right)}}}} + {\sum\limits_{m}{\sum\limits_{n}{\beta_{m}\alpha_{n}^{*}\frac{\partial{R_{h}\left( {x_{m} - x_{n}} \right)}}{\partial x_{m}}}}} - {\sum\limits_{m}{\beta_{n}^{*}\frac{\partial{R_{h}\left( {y - x_{n}} \right)}}{\partial x_{n}}}} + {\sum\limits_{m}{\sum\limits_{n}{\alpha_{m}\beta_{m}\frac{\partial{R_{h}\left( {x_{m} - x_{n}} \right)}}{\partial x_{n}}}}} + {\sum\limits_{m}{\sum\limits_{n}{\beta_{m}\beta_{n}^{*}\frac{\partial{R_{h}\left( {x_{m} - x_{n}} \right)}^{\prime}}{{\partial x_{m}}{\partial x_{n}}}}}}} = {{R_{hh}(0)} - {g^{H}\theta} - {\theta^{H}g} + {\theta^{H}{\prod\theta}}}}} & {{Eq}.\mspace{14mu} 11}\end{matrix}$

In Eq. 11, the parameters θ, g and Π are the following:

$\begin{matrix}{{\theta = \begin{bmatrix}a \\b\end{bmatrix}},} & {{Eq}.\mspace{14mu} 12} \\{{g = \begin{bmatrix}r \\{- r^{\prime}}\end{bmatrix}},{and}} & {{Eq}.\mspace{14mu} 13} \\{{\prod{= \begin{bmatrix}R & R^{\prime} \\{- R^{\prime}} & {- R^{''}}\end{bmatrix}}},} & {{Eq}.\mspace{14mu} 14}\end{matrix}$

where the parameters in Eqs. 12, 13 and 14 may be represented asfollows:

a _(m)=α_(m),  Eq. 15

b _(m)=β_(m),  Eq. 16

r _(m) =R _(hh)(y−x _(m)),  Eq. 17

R _(mn) =R _(hh)(x _(m) −x _(n)),  Eq. 18

R′ _(mn) =R′ _(hh)(x _(m) −x _(n)), and  Eq. 19

R″ _(mn) =R″ _(hh)(x _(m) −x _(n)).  Eq. 20

The calculated interpolation parameters α_(m) and β_(m) may be found byminimizing the variance of the interpolation error by solving thefollowing set of normal equations:

Πθ=g.  Eq. 21

Referring to FIG. 2, thus, to summarize, a technique 120 may beperformed in accordance with embodiments of the invention for purposesof interpolating seismic measurements from actual seismic measurementsthat were acquired at non-uniformly spaced sensor locations. Pursuant tothe technique 120, the interpolated seismic measurements are modeled(block 124) as a random process that is characterized by the acquiredseismic measurements and an interpolation error. The technique 120includes minimizing the interpolation error, pursuant to block 128, anddetermining (block 132) the interpolated seismic measurements based atleast in part on the minimization of the interpolation error.

As described above, a more specific technique 140 may be used inaccordance with some embodiments of the invention for purposes ofdetermining the interpolated seismic measurements. The technique 140includes modeling (block 144) the interpolated seismic measurements as arandom process that is characterized by the acquired seismic data and aninterpolation error. The acquired seismic measurements areautocorrelated, pursuant to block 148, and the interpolation error isminimized based on the results of the autocorrelation, pursuant to block152. The interpolated seismic measurements may then be determined,pursuant to block 156, based at least in part on the minimization of theinterpolation error.

It is noted that the interpolation may be performed in the frequency (f)and space (x,y,z) domain. For determination in the frequency domain, themeasurements may be divided among frequency bands such thatinterpolation is performed for each frequency band independently. Eachfrequency band may have a different associated spatial bandwidth.

The autocorrelation of the acquired seismic measurements may beperformed in a number of different ways, depending on the particularembodiment of the invention. Referring to FIG. 4, in accordance withsome embodiments of the invention, a technique 170 includes determininga power density spectrum of the acquired seismic measurements, pursuantto block 172 and performing an inverse Fourier transform of thedetermined power density spectrum, pursuant to block 176, for purposesof performing the autocorrelation.

In some applications, the power density spectrum may not be knownexactly. Therefore, the spectrum may be estimated from the acquiredseismic measurements by using spectrum estimation algorithms, as can beappreciated by one of skill in the art. Referring to FIG. 5, inaccordance with some embodiments of the invention, a technique 180 maybe used for purposes of estimating the power density spectrum. Pursuantto the technique 180, a power density spectrum, which was previouslyattained at a lower frequency is obtained, pursuant to block 184. Inthis regard, the spectrum may have already been computed at a lowerfrequency. The technique 180 includes estimating (block 188) the powerdensity spectrum at a different frequency based at least in part on thespectrum that was already determined at the lower frequency.

Referring to FIG. 6, in accordance with other embodiments of theinvention, in some applications where the power density spectrum is notexactly known, the spectrum may be modeled by using prior knowledgeabout the spectrum. In this regard, a technique 200 includes obtainingprior knowledge about the power density spectrum, pursuant to block 204and estimating (block 210) the power density spectrum based on theobtained knowledge. As a more specific example, if it is known that theacquired seismic data measurements are band-limited, then a model forthe autocorrelation function may be described, in its simplest form, asfollows:

R _(hh)(x)=K sin c(σx),  Eq. 22

where “sin c(σx)” represents the sin c function, “K” represents aconstant and “σ” represents twice the Nyquist rate of the bandwidth ofthe measured seismic data.

In some applications, the seismic data measurements may only includepressure data and not particle motion data. In these applications, theβ_(m) coefficient is set equal to zero, and the α_(m) is determined asthe solution of the following normal equation:

Ra=r.  Eq. 23

If Eq. 21 is over-determined, then a minimum norm solution may beobtained by determining the singular value decomposition of Π. Asanother example, if Eq. 21 is over-determined, a regularized solutionmay be obtained by using Tikhonov regularization.

It is noted that the model of the power density spectrum need not beband-limited. Therefore, in the presence of sufficient information aboutthe signal's spectrum, the power density spectrum estimation algorithmmay not be limited by the Nyquist rate.

As an example of another embodiment of the invention, in the presence ofnoise, the statistics of the noise may be incorporated into theinterpolator. Other variations are contemplated and are within the scopeof the appended claims.

Referring to FIG. 7, in accordance with some embodiments of theinvention, a seismic data processing system 320 may perform at leastpart of the techniques that are disclosed herein for purposesinterpolating seismic measurements (pressure and particle motionmeasurements, as example) at uniformly spaced locations based on seismicmeasurements that were acquired at non-uniformly spaced sensorlocations. In accordance with some embodiments of the invention, thesystem 320 may include a processor 350, such as one or moremicroprocessors and/or microcontrollers. The processor 350 may belocated on a streamer 30 (FIG. 1), located on the vessel 20 or locatedat a land-based processing facility (as examples), depending on theparticular embodiment of the invention.

The processor 350 may be coupled to a communication interface 360 forpurposes of receiving seismic data that corresponds to the acquiredpressure and/or particle motion measurements. Thus, in accordance withembodiments of the invention described herein, the processor 350, whenexecuting instructions stored in a memory of the seismic data processingsystem 320, may receive multi-component data that is acquired bymulti-component seismic sensors while in tow. It is noted that,depending on the particular embodiment of the invention, themulti-component data may be data that is directly received from themulti-component seismic sensor as the data is being acquired (for thecase in which the processor 350 is part of the survey system, such aspart of the vessel or streamer) or may be multi-component data that waspreviously acquired by the seismic sensors while in tow and stored andcommunicated to the processor 350, which may be in a land-basedfacility, for example.

As examples, the interface 360 may be a USB serial bus interface, anetwork interface, a removable media (such as a flash card, CD-ROM,etc.) interface or a magnetic storage interface (IDE or SCSI interfaces,as examples). Thus, the interface 360 may take on numerous forms,depending on the particular embodiment of the invention.

In accordance with some embodiments of the invention, the interface 360may be coupled to a memory 340 of the seismic data processing system 320and may store, for example, various input and/or output data setsinvolved with the techniques 120, 140, 170, 180 and/or 200, as indicatedby reference numeral 348. The memory 340 may store program instructions344, which when executed by the processor 350, may cause the processor350 to perform one or more of the techniques that are disclosed herein,such as the techniques 120, 140, 170, 108 and/or 200 and display resultsobtained via the technique(s) on a display (not shown in FIG. 7) of thesystem 320, in accordance with some embodiments of the invention.

Many variations are contemplated and are within the scope of theappended claims. For example, in accordance with embodiments of theinvention, the streamers may be in any configuration, such as anover/under configuration, a three-dimensional spread, etc. Furthermore,in accordance with other embodiments of the invention, the seismicsensors may be disposed in a sensor cable, such as an ocean bottomcable.

While the present invention has been described with respect to a limitednumber of embodiments, those skilled in the art, having the benefit ofthis disclosure, will appreciate numerous modifications and variationstherefrom. It is intended that the appended claims cover all suchmodifications and variations as fall within the true spirit and scope ofthis present invention.

What is claimed is:
 1. A method comprising: modeling interpolatedseismic measurements as a random process, wherein the seismicmeasurements comprise pressure measurements or particle motionmeasurements that are acquired at sensor locations, and an interpolationerror; processing the seismic measurements in a processor-based machineto determine the interpolated seismic measurements based at least inpart on a minimization of the interpolation error; and: further modelingthe interpolated seismic measurements based on the seismic measurementsand the interpolation error.
 2. The method of claim 1, wherein thesensor locations comprise non-co-located sensor locations.
 3. The methodof claim 1, wherein processing the seismic measurements comprisesdetermining the interpolated seismic measurements at locationsassociated with a uniform sampling.
 4. The method of claim 1, whereinthe sensor locations comprise locations associated with a non-uniformsampling.
 5. The method of claim 1, wherein the act of minimizing theinterpolation error comprises minimizing a variation of theinterpolation error.
 6. The method of claim 1, wherein processing theseismic measurements comprises calculating autocorrelation of theseismic measurements at the sensor locations.
 7. The method of claim 6,wherein the act of calculating the autocorrelation comprises determininga power spectral density of the seismic measurements acquired at thesensor locations.
 8. The method of claim 7, wherein the autocorrelationof seismic measurements is computed as the inverse Fourier transform ofan a priori model of the power spectral density of the seismicmeasurements.
 9. The method of claim 7, wherein the power spectraldensity of the seismic measurements is modeled as a constant forband-limited measurements.
 10. The method of claim 7, wherein the act ofdetermining the power spectral density comprises determining the powerspectral density at a first frequency based on a power spectral densitydetermined at a different second frequency.
 11. The method of claim 1further comprising: dividing the measurements among frequency bands; andperforming the interpolation for each frequency band independently. 12.The method of claim 11, wherein each frequency band has a differentassociated spatial bandwidth.
 13. The method of claim 1, wherein themeasurements are acquired in a two-dimensional (2D) spatial plane. 14.The method of claim 1, wherein the measurements are acquired in athree-dimensional (3D) spatial plane.
 15. The method of claim 12,wherein the different spatial bandwidths are determined by the speed ofpropagation of the signals.
 16. The method of claim 1, wherein theinterpolated seismic measurements comprise three particle motioncomponents and a pressure component.
 17. An article comprisinginstructions stored on a computer accessible storage medium that whenexecuted by a processor-based system cause the processor-based systemto: model interpolated seismic measurements as a random processcharacterized by seismic data measurements acquired at sensor locationsand an interpolation error; and determine the interpolated seismicmeasurements based at least in part on a minimization of theinterpolation error.
 18. The article of claim 17, the storage mediumstoring instructions to cause the processor-based system to minimize avariation of the interpolation error to minimize the interpolationerror.
 19. The article of claim 17, the storage medium storinginstructions to cause the processor to further model the interpolatedseismic measurements as statistics of the seismic measurements and theinterpolation error.
 20. The article of claim 17, wherein the seismicmeasurements acquired at the sensor locations comprise pressure data andparticle motion data acquired at sensor locations along at least onestreamer.
 21. The article of claim 17, the storage medium storingcomprising instructions to cause the processor-based system to performautocorrelation of the seismic measurements acquired at the sensorlocations.
 22. The article of claim 17, wherein the sensor locations areassociated with a non-uniform sampling, and the instructions whenexecuted cause the processor-based system to determine interpolatedmeasurements associated with a uniform sampling.
 23. A systemcomprising: an interface to receive a set of seismic measurementsacquired at non-uniformly spaced sensor locations; and a processor tomodel interpolated seismic measurements at uniformly spaced locations asa random process characterized by the seismic data measurements acquiredat the non-uniformly spaced sensor locations, an interpolation error,and statistics of the seismic measurements and interpolation error; anddetermine the interpolated seismic measurements at the uniformly spacedlocations based at least in part on a minimization of the interpolationerror.
 24. The system of claim 23, wherein the processor is adapted tofurther model the interpolated seismic measurements as statistics of theseismic measurements and the interpolation error.
 25. The system ofclaim 23, wherein the processor is adapted to minimize a variation ofthe interpolation error.
 26. The system of claim 23, wherein the seismicmeasurements acquired at the non-uniformly spaced sensor locationscomprise pressure data and particle motion data acquired at sensorlocations of at least one streamer.
 27. The system of claim 23, whereinthe processor is adapted to perform autocorrelation of the set ofseismic data measurements.
 28. The system of claim 27, wherein theprocessor is adapted to perform an inverse Fourier transform on a powerspectral density of the seismic measurements acquired at thenon-uniformly spaced sensor locations to compute autocorrelation of theseismic data measurements.
 29. The system of claim 28, furthercomprising: a survey vessel to tow at least one sensor cable to acquiresaid seismic measurements acquired at the non-uniformly spaced sensorlocations, wherein the processor is located on said at least one sensorcable.
 30. The system of claim 29, wherein said at least one sensorcable comprises an ocean bottom cable or a streamer.
 31. The system ofclaim 23, further comprising: a survey vessel to tow at least onestreamer to acquire said seismic measurements acquired at thenon-uniformly spaced sensor locations, wherein the processor is notlocated on said at least one streamer.